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Linear Algebra Section 4.4 Flashcards
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Linear Algebra Section 4.4 Flashcards

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Questions and Answers

What is a linear combination of vectors?

  • A set of vectors cannot be combined.
  • A vector can be expressed as a sum of scaled vectors. (correct)
  • A vector is always independent from others.
  • A vector can only be created from itself.
  • What is a spanning set of a vector space?

    A set where every vector in the space can be represented as a linear combination of its vectors.

    Define the span of a set.

    The set of all linear combinations of the vectors in the set.

    Span(S) is a subspace of V.

    <p>True</p> Signup and view all the answers

    When is a set of vectors called linearly independent?

    <p>When the only solution to the vector equation is the trivial solution.</p> Signup and view all the answers

    A set S is called linearly dependent if it has _____ solutions.

    <p>nontrivial</p> Signup and view all the answers

    What are the steps to test for linear independence?

    <p>Create a system of equations from the vector equation and check if it has a unique solution.</p> Signup and view all the answers

    A set is linearly independent if one of the vectors is a linear combination of others.

    <p>False</p> Signup and view all the answers

    Two vectors in a vector space are linearly independent if one is a scalar multiple of the other.

    <p>False</p> Signup and view all the answers

    Study Notes

    Linear Combinations

    • A vector v in a vector space V is a linear combination of vectors u1, u2, ..., uk if it can be expressed as c1u1 + c2u2 + ... + ckuk, where c1, c2, ..., ck are scalars.

    Spanning Sets

    • A subset S = {v1, v2, ..., vk} of a vector space V is a spanning set if every vector in V can be expressed as a linear combination of vectors in S.
    • If S spans V, it means that the entirety of V can be reached by taking linear combinations of vectors in S.

    Span of a Set

    • The span of a set of vectors S = {v1, v2, ..., vk} is defined as the collection of all linear combinations of the vectors in S.
    • Denoted as span(S) or span{v1, v2, ..., vk}, it represents all possible linear combinations using real numbers as coefficients.
    • If span(S) = V, then it is stated that V is spanned by S.

    Subspace Properties

    • Span(S) is a subspace of the vector space V that includes all linear combinations of vectors in S.
    • It is the smallest subspace containing S, meaning any other subspace that contains S must also include span(S).

    Linear Dependence and Independence

    • A set of vectors S = {v1, v2, ..., vk} is linearly independent if the equation c1v1 + c2v2 + ... + ckvk = 0 only holds for the trivial solution c1 = 0, c2 = 0, ..., ck = 0.
    • If there are nontrivial solutions (other than the trivial one), then the set S is linearly dependent.

    Testing Dependence/Independence

    • To test whether a set S is linearly independent or dependent:
      • Set up the equation c1v1 + c2v2 + ... + ckvk = 0 and transform it into a system of linear equations.
      • Apply Gaussian elimination to assess the uniqueness of the solution.
      • If the solution is unique (only trivial), S is independent; if there are nontrivial solutions, it is dependent.

    Theorem on Linear Independence

    • A set of vectors S = {v1, v2, ..., vk}, with k ≥ 2, is linearly independent only if at least one vector can be expressed as a linear combination of the others.

    Corollary on Two Vectors

    • Two vectors u and v in a vector space V are linearly independent if and only if neither is a scalar multiple of the other.

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    Test your knowledge on spanning sets and linear independence with these flashcards. Understand key concepts like linear combinations and spanning sets in vector spaces. Perfect for students studying linear algebra.

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